Canonical Form - Definition

Definition

Suppose we have some set S of objects, with an equivalence relation. A canonical form is given by designating some objects of S to be "in canonical form", such that every object under consideration is equivalent to exactly one object in canonical form. In other words, the canonical forms in S represent the equivalence classes, once and only once. To test whether two objects are equivalent, it then suffices to test their canonical forms for equality. A canonical form thus provides a classification theorem and more, in that it not just classifies every class, but gives a distinguished (canonical) representative.

In practical terms, one wants to be able to recognize the canonical forms. There is also a practical, algorithmic question to consider: how to pass from a given object s in S to its canonical form s*? Canonical forms are generally used to make operating with equivalence classes more effective. For example in modular arithmetic, the canonical form for a residue class is usually taken as the least non-negative integer in it. Operations on classes are carried out by combining these representatives and then reducing the result to its least non-negative residue. The uniqueness requirement is sometimes relaxed, allowing the forms to be unique up to some finer equivalence relation, like allowing reordering of terms (if there is no natural ordering on terms).

A canonical form may simply be a convention, or a deep theorem.

For example, polynomials are conventionally written with the terms in descending powers: it is more usual to write x2 + x + 30 than x + 30 + x2, although the two forms define the same polynomial. By contrast, the existence of Jordan canonical form for a matrix is a deep theorem.

Read more about this topic:  Canonical Form

Famous quotes containing the word definition:

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.
    Elaine Heffner (20th century)

    It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.
    François, Duc De La Rochefoucauld (1613–1680)